Integrand size = 23, antiderivative size = 93 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {b^2 c}{a^4 x}+\frac {b^2 d \log (x)}{a^4}-\frac {b^2 (b c+a d) \log (a-b x)}{2 a^5}+\frac {b^2 (b c-a d) \log (a+b x)}{2 a^5} \] Output:
-1/3*c/a^2/x^3-1/2*d/a^2/x^2-b^2*c/a^4/x+b^2*d*ln(x)/a^4-1/2*b^2*(a*d+b*c) *ln(-b*x+a)/a^5+1/2*b^2*(-a*d+b*c)*ln(b*x+a)/a^5
Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.90 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}-\frac {b^2 c}{a^4 x}+\frac {b^3 c \text {arctanh}\left (\frac {b x}{a}\right )}{a^5}+\frac {b^2 d \log (x)}{a^4}-\frac {b^2 d \log \left (a^2-b^2 x^2\right )}{2 a^4} \] Input:
Integrate[(c + d*x)/(x^4*(a^2 - b^2*x^2)),x]
Output:
-1/3*c/(a^2*x^3) - d/(2*a^2*x^2) - (b^2*c)/(a^4*x) + (b^3*c*ArcTanh[(b*x)/ a])/a^5 + (b^2*d*Log[x])/a^4 - (b^2*d*Log[a^2 - b^2*x^2])/(2*a^4)
Time = 0.42 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {523, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx\) |
\(\Big \downarrow \) 523 |
\(\displaystyle \int \left (\frac {b^3 (a d+b c)}{2 a^5 (a-b x)}-\frac {b^3 (a d-b c)}{2 a^5 (a+b x)}+\frac {b^2 c}{a^4 x^2}+\frac {b^2 d}{a^4 x}+\frac {c}{a^2 x^4}+\frac {d}{a^2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {b^2 (a d+b c) \log (a-b x)}{2 a^5}+\frac {b^2 (b c-a d) \log (a+b x)}{2 a^5}-\frac {b^2 c}{a^4 x}+\frac {b^2 d \log (x)}{a^4}-\frac {c}{3 a^2 x^3}-\frac {d}{2 a^2 x^2}\) |
Input:
Int[(c + d*x)/(x^4*(a^2 - b^2*x^2)),x]
Output:
-1/3*c/(a^2*x^3) - d/(2*a^2*x^2) - (b^2*c)/(a^4*x) + (b^2*d*Log[x])/a^4 - (b^2*(b*c + a*d)*Log[a - b*x])/(2*a^5) + (b^2*(b*c - a*d)*Log[a + b*x])/(2 *a^5)
Int[((x_)^(m_.)*((c_) + (d_.)*(x_)))/((a_) + (b_.)*(x_)^2), x_Symbol] :> In t[ExpandIntegrand[x^m*((c + d*x)/(a + b*x^2)), x], x] /; FreeQ[{a, b, c, d} , x] && IntegerQ[m]
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\left (a d -b c \right ) b^{2} \ln \left (b x +a \right )}{2 a^{5}}-\frac {c}{3 a^{2} x^{3}}-\frac {d}{2 a^{2} x^{2}}-\frac {b^{2} c}{a^{4} x}+\frac {b^{2} d \ln \left (x \right )}{a^{4}}-\frac {b^{2} \left (a d +b c \right ) \ln \left (-b x +a \right )}{2 a^{5}}\) | \(86\) |
norman | \(\frac {-\frac {c}{3 a^{2}}-\frac {d x}{2 a^{2}}-\frac {b^{2} x^{2} c}{a^{4}}}{x^{3}}+\frac {b^{2} d \ln \left (x \right )}{a^{4}}-\frac {\left (a d -b c \right ) b^{2} \ln \left (b x +a \right )}{2 a^{5}}-\frac {b^{2} \left (a d +b c \right ) \ln \left (-b x +a \right )}{2 a^{5}}\) | \(86\) |
parallelrisch | \(\frac {6 d \,b^{2} \ln \left (x \right ) a \,x^{3}-3 \ln \left (b x -a \right ) x^{3} a \,b^{2} d -3 \ln \left (b x -a \right ) x^{3} b^{3} c -3 \ln \left (b x +a \right ) x^{3} a \,b^{2} d +3 \ln \left (b x +a \right ) x^{3} b^{3} c -6 a \,b^{2} c \,x^{2}-3 a^{3} d x -2 c \,a^{3}}{6 a^{5} x^{3}}\) | \(111\) |
risch | \(\frac {-\frac {c}{3 a^{2}}-\frac {d x}{2 a^{2}}-\frac {b^{2} x^{2} c}{a^{4}}}{x^{3}}-\frac {b^{2} \ln \left (b x -a \right ) d}{2 a^{4}}-\frac {b^{3} \ln \left (b x -a \right ) c}{2 a^{5}}-\frac {b^{2} \ln \left (-b x -a \right ) d}{2 a^{4}}+\frac {b^{3} \ln \left (-b x -a \right ) c}{2 a^{5}}+\frac {b^{2} d \ln \left (x \right )}{a^{4}}\) | \(112\) |
Input:
int((d*x+c)/x^4/(-b^2*x^2+a^2),x,method=_RETURNVERBOSE)
Output:
-1/2*(a*d-b*c)*b^2/a^5*ln(b*x+a)-1/3*c/a^2/x^3-1/2*d/a^2/x^2-b^2*c/a^4/x+b ^2*d*ln(x)/a^4-1/2*b^2*(a*d+b*c)*ln(-b*x+a)/a^5
Time = 0.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {6 \, a b^{2} d x^{3} \log \left (x\right ) - 6 \, a b^{2} c x^{2} - 3 \, a^{3} d x + 3 \, {\left (b^{3} c - a b^{2} d\right )} x^{3} \log \left (b x + a\right ) - 3 \, {\left (b^{3} c + a b^{2} d\right )} x^{3} \log \left (b x - a\right ) - 2 \, a^{3} c}{6 \, a^{5} x^{3}} \] Input:
integrate((d*x+c)/x^4/(-b^2*x^2+a^2),x, algorithm="fricas")
Output:
1/6*(6*a*b^2*d*x^3*log(x) - 6*a*b^2*c*x^2 - 3*a^3*d*x + 3*(b^3*c - a*b^2*d )*x^3*log(b*x + a) - 3*(b^3*c + a*b^2*d)*x^3*log(b*x - a) - 2*a^3*c)/(a^5* x^3)
Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (85) = 170\).
Time = 0.97 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.00 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {b^{2} d \log {\left (x \right )}}{a^{4}} - \frac {2 a^{2} c + 3 a^{2} d x + 6 b^{2} c x^{2}}{6 a^{4} x^{3}} - \frac {b^{2} \left (a d - b c\right ) \log {\left (x + \frac {6 a^{4} b^{4} d^{3} - 3 a^{3} b^{4} d^{2} \left (a d - b c\right ) + 2 a^{2} b^{6} c^{2} d - 3 a^{2} b^{4} d \left (a d - b c\right )^{2} + a b^{6} c^{2} \left (a d - b c\right )}{9 a^{2} b^{6} c d^{2} - b^{8} c^{3}} \right )}}{2 a^{5}} - \frac {b^{2} \left (a d + b c\right ) \log {\left (x + \frac {6 a^{4} b^{4} d^{3} - 3 a^{3} b^{4} d^{2} \left (a d + b c\right ) + 2 a^{2} b^{6} c^{2} d - 3 a^{2} b^{4} d \left (a d + b c\right )^{2} + a b^{6} c^{2} \left (a d + b c\right )}{9 a^{2} b^{6} c d^{2} - b^{8} c^{3}} \right )}}{2 a^{5}} \] Input:
integrate((d*x+c)/x**4/(-b**2*x**2+a**2),x)
Output:
b**2*d*log(x)/a**4 - (2*a**2*c + 3*a**2*d*x + 6*b**2*c*x**2)/(6*a**4*x**3) - b**2*(a*d - b*c)*log(x + (6*a**4*b**4*d**3 - 3*a**3*b**4*d**2*(a*d - b* c) + 2*a**2*b**6*c**2*d - 3*a**2*b**4*d*(a*d - b*c)**2 + a*b**6*c**2*(a*d - b*c))/(9*a**2*b**6*c*d**2 - b**8*c**3))/(2*a**5) - b**2*(a*d + b*c)*log( x + (6*a**4*b**4*d**3 - 3*a**3*b**4*d**2*(a*d + b*c) + 2*a**2*b**6*c**2*d - 3*a**2*b**4*d*(a*d + b*c)**2 + a*b**6*c**2*(a*d + b*c))/(9*a**2*b**6*c*d **2 - b**8*c**3))/(2*a**5)
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {b^{2} d \log \left (x\right )}{a^{4}} + \frac {{\left (b^{3} c - a b^{2} d\right )} \log \left (b x + a\right )}{2 \, a^{5}} - \frac {{\left (b^{3} c + a b^{2} d\right )} \log \left (b x - a\right )}{2 \, a^{5}} - \frac {6 \, b^{2} c x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, a^{4} x^{3}} \] Input:
integrate((d*x+c)/x^4/(-b^2*x^2+a^2),x, algorithm="maxima")
Output:
b^2*d*log(x)/a^4 + 1/2*(b^3*c - a*b^2*d)*log(b*x + a)/a^5 - 1/2*(b^3*c + a *b^2*d)*log(b*x - a)/a^5 - 1/6*(6*b^2*c*x^2 + 3*a^2*d*x + 2*a^2*c)/(a^4*x^ 3)
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {b^{2} d \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} c - a b^{3} d\right )} \log \left ({\left | b x + a \right |}\right )}{2 \, a^{5} b} - \frac {{\left (b^{4} c + a b^{3} d\right )} \log \left ({\left | b x - a \right |}\right )}{2 \, a^{5} b} - \frac {6 \, b^{2} c x^{2} + 3 \, a^{2} d x + 2 \, a^{2} c}{6 \, a^{4} x^{3}} \] Input:
integrate((d*x+c)/x^4/(-b^2*x^2+a^2),x, algorithm="giac")
Output:
b^2*d*log(abs(x))/a^4 + 1/2*(b^4*c - a*b^3*d)*log(abs(b*x + a))/(a^5*b) - 1/2*(b^4*c + a*b^3*d)*log(abs(b*x - a))/(a^5*b) - 1/6*(6*b^2*c*x^2 + 3*a^2 *d*x + 2*a^2*c)/(a^4*x^3)
Time = 0.13 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {\ln \left (a+b\,x\right )\,\left (b^3\,c-a\,b^2\,d\right )}{2\,a^5}-\frac {\frac {c}{3\,a^2}+\frac {d\,x}{2\,a^2}+\frac {b^2\,c\,x^2}{a^4}}{x^3}-\frac {\ln \left (a-b\,x\right )\,\left (c\,b^3+a\,d\,b^2\right )}{2\,a^5}+\frac {b^2\,d\,\ln \left (x\right )}{a^4} \] Input:
int((c + d*x)/(x^4*(a^2 - b^2*x^2)),x)
Output:
(log(a + b*x)*(b^3*c - a*b^2*d))/(2*a^5) - (c/(3*a^2) + (d*x)/(2*a^2) + (b ^2*c*x^2)/a^4)/x^3 - (log(a - b*x)*(b^3*c + a*b^2*d))/(2*a^5) + (b^2*d*log (x))/a^4
Time = 0.19 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.23 \[ \int \frac {c+d x}{x^4 \left (a^2-b^2 x^2\right )} \, dx=\frac {-3 \,\mathrm {log}\left (-b x -a \right ) a \,b^{2} d \,x^{3}+3 \,\mathrm {log}\left (-b x -a \right ) b^{3} c \,x^{3}-3 \,\mathrm {log}\left (-b x +a \right ) a \,b^{2} d \,x^{3}-3 \,\mathrm {log}\left (-b x +a \right ) b^{3} c \,x^{3}+6 \,\mathrm {log}\left (x \right ) a \,b^{2} d \,x^{3}-2 a^{3} c -3 a^{3} d x -6 a \,b^{2} c \,x^{2}}{6 a^{5} x^{3}} \] Input:
int((d*x+c)/x^4/(-b^2*x^2+a^2),x)
Output:
( - 3*log( - a - b*x)*a*b**2*d*x**3 + 3*log( - a - b*x)*b**3*c*x**3 - 3*lo g(a - b*x)*a*b**2*d*x**3 - 3*log(a - b*x)*b**3*c*x**3 + 6*log(x)*a*b**2*d* x**3 - 2*a**3*c - 3*a**3*d*x - 6*a*b**2*c*x**2)/(6*a**5*x**3)